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To simplify means to write an expression in a shorter, equal form. You combine like terms, multiply out brackets, and cancel common factors in fractions. For example, \(\frac{x^2 – 1}{x – 1} = x + 1\) for every \(x \ne 1\).
How to Use the Simplify Calculator
- Type an expression, such as
(x+1)^2 - (x-1)^2or(x^2-1)/(x-1). - Press Solve.
- You get the simplest form. When it helps, you also see the expanded form and the factored form, so you can pick the one your teacher wants.
The Main Simplifying Rules
| Rule | Example |
|---|---|
| Combine like terms | \(2x + 3x – 4 = 5x – 4\) |
| Distribute | \(3(x + 2) = 3x + 6\) |
| Square a binomial | \((x + 1)^2 = x^2 + 2x + 1\) |
| Cancel common factors | \(\frac{x^2 + 5x + 6}{x + 2} = x + 3\) |
| Exponent rules | \(x^2 \cdot x^3 = x^5\), \(\frac{x^5}{x^2} = x^3\) |
Worked Examples
Example 1: \(2x + 3x – 4 + x^2\)
The like terms are \(2x\) and \(3x\), which add to \(5x\). Write the answer in standard form (highest power first): \(x^2 + 5x – 4\).
Example 2: \((x + 1)^2 – (x – 1)^2\)
Expand each square: \((x^2 + 2x + 1) – (x^2 – 2x + 1)\). The \(x^2\) terms and the 1s cancel, leaving \(4x\).
Example 3: \(\frac{x^2 – 1}{x – 1}\)
Factor the top: \(x^2 – 1 = (x – 1)(x + 1)\). Cancel \((x – 1)\) to get \(x + 1\). Note that the original fraction is still undefined at \(x = 1\).
Example 4: \(x(x + 2) – 3(x + 2)\)
Expand: \(x^2 + 2x – 3x – 6 = x^2 – x – 6\). Factored, that is \((x – 3)(x + 2)\), because \((x + 2)\) was a common factor from the start.
Why Simplifying Matters in Calculus
A simpler expression is easier to differentiate and integrate. Before you use the power rule, rewrite \(\frac{x^3 + x}{x}\) as \(x^2 + 1\). Before you find a limit, cancel common factors. Try both with the derivative calculator and the limit calculator.
Common Mistakes
- Cancelling terms instead of factors. In \(\frac{x + 2}{x}\), you cannot cancel the \(x\). Only whole factors cancel.
- Squaring term by term. \((x + 1)^2 \ne x^2 + 1\). The middle term \(2x\) is easy to forget.
- Dropping a negative sign. \(-(x – 1) = -x + 1\).
Learn the Math Behind the Simplify Calculator
Many integrals and limits get easy after one algebra step. Simplifying first is often the fastest path to the answer.
Frequently Asked Questions
Which form is the “simplest”?
It depends on the job. Expanded form is best for adding and integrating. Factored form is best for solving equations and cancelling. This calculator shows both when they differ.
How do I know the answer is equal to my expression?
The calculator plugs several numbers into both expressions and checks that they give the same values. That is what the ✓ Verified badge means.
Can it simplify fractions with polynomials?
Yes. It finds the greatest common factor of the top and bottom and cancels it.